Proof. 
It may be seen that \(H\)  contains a derangement of order \(p\) ,
in fact a semiregular element \(\alpha \)  of order \(p\)  such that
the set of orbits of \(\langle \alpha \rangle \)  forms an \(H\) -invariant partition \({\cal {B}}\) ,
see [1]}. Let \(\bar{H}\)  be the permutation group
induced by the action of \(H\)  on \({\cal {B}}\) .
Then, by [2]}, \(\rho (H)\le \rho (\bar{H})\) .
Since \(\bar{H}\)  is a transitive group of prime degree we have
\(\rho (\bar{H})=1\)  by [2]},
and so \(\rho (H)=1\) .
Moreover, by [4]},
\(\rho (G)\le \rho (H)\) , and the result follows.
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